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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Analytic function ： ウィキペディア英語版 Analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. == Definitions == Formally, a function ƒ is ''real analytic'' on an open set ''D'' in the real line if for any ''x''0 in ''D'' one can write :$$ f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients ''a''0, ''a''1, ... are real numbers and the series is convergent to ƒ(''x'') for ''x'' in a neighborhood of ''x''0. Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point ''x''0 in its domain :$T(x)\; =\; \backslash sum\_^\; \backslash frac\; (x-x\_0)^$ converges to f(''x'') for ''x'' in a neighborhood of ''x''0 pointwise (and locally uniformly). The set of all real analytic functions on a given set ''D'' is often denoted by ''Cω''(''D''). A function ƒ defined on some subset of the real line is said to be real analytic at a point ''x'' if there is a neighborhood ''D'' of ''x'' on which ƒ is real analytic. The definition of a ''complex analytic function'' is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function is complex analytic if and only if it is holomorphic i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions.〔"A function ''f'' of the complex variable ''z'' is ''analytic'' at point z0 if its derivative exists not only at z but at each point ''z'' in some neighborhood of z0. It is analytic in a region ''R'' if it is analytic as every point in ''R''. The term ''holomorphic'' is also used in the literature do denote analyticity." Churchill, Brown, and Verhey ''Complex Variables and Applications'' McGraw-Hill 1948 ISBN 0-07-010855-2 pg 46〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Analytic function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.